The rule fits in just two lines. If the number is even, divide it by 2; if odd, triple it and add 1. Keep repeating this, and the Collatz conjecture says every number you start with eventually reaches 1. The rule is this simple, yet no one has been able to prove it since it was first posed in 1937.
Pick a number. If it's even, divide it by 2; if it's odd, triple it and add 1. Apply the same rule again to whatever new number comes out. Keep repeating this process, and every single number checked by computer so far has eventually landed in a loop, circling 4 → 2 → 1.
The question is: "does this really hold for every natural number?" No matter how large a number you feed in, no matter how wildly it seems to bounce around, it has always ended at 1 so far — but there's still no proof that it's "always true for every number."
Pick a starting number and use the slider to follow it one step at a time. Even numbers are shown in blue, odd numbers in coral.
Depending on the starting number, both the number of steps to reach 1 and the highest value reached along the way vary wildly. Even a small number like 27 takes as many as 111 steps and climbs as high as 9232.
This problem, first posed in 1937 by the German mathematician Lothar Collatz, is especially famous because the rule is simple enough for an elementary school student to understand. Computers have checked every number smaller than 2⁶⁸ (about 3×10²⁰), and not a single exception has been found.
But because the rule works completely differently depending on whether a number is even or odd, there's no good mathematical theory that handles a number growing and shrinking all at once. The famous Hungarian mathematician Paul Erdős went so far as to say that mathematics simply isn't ready to tackle this problem yet. It also carries a prize, so mathematicians and amateurs around the world are still taking it on today.